#4430402

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(14,16)(18,19)(21,22,24,26)(23,27,28,25);(5,7,9,8)(6,10,11,12)(13,14,17,16)(15,19,20,18)(23,28)(25,27);(1,2)(3,4)(13,15)(14,18)(16,19)(17,20)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20);(5,8,9,7)(6,10,11,12)(13,16,17,14)(15,19,20,18)(21,22)(23,25)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4337598);